A novel kernelized angle metric for state similarity in nonlinear dynamic systems

Accurate state similarity measurement is critical for predicting nonlinear dynamical systems, yet traditional metrics like Euclidean distance often fail by ignoring the system's intrinsic manifold geometry. While manifold-based distances offer an alternative, they can be computationally expensive and sensitive to data imperfections. To address these challenges, we introduce the Kernelized Angle Metric (KAM), a novel, computationally efficient measure for state-space similarity. KAM is designed as a flexible framework that adaptively combines two functionally specialized components: (1) a distance term that non-linearly scales Euclidean distance to reflect the characteristic exponential divergence of trajectories often found in chaotic systems, and (2) a robust angle term based on cosine similarity to assess directional coherence. A weighting parameter balances the contribution of these two components, allowing the metric to be optimized for specific data characteristics. Through systematic evaluation on multiple datasets within the Empirical Dynamic Modeling (EDM) framework, we demonstrate that KAM-based methods consistently outperform Euclidean-based benchmarks, particularly showing robustness in noisy conditions. More profoundly, the results reveal a critical algorithm-metric coupling: KAM provides a substantial performance leap for simpler algorithms like Simplex, while its performance is competitive with more computationally intensive manifold distance methods when used with complex algorithms like S-map. This positions KAM as a powerful and practical tool that excels at empowering simpler models and offers an excellent balance of accuracy, robustness, and computational efficiency for analyzing complex systems.